A bandgap voltage reference circuit is based on addition of two voltages having equal and opposite temperature coefficient. The first voltage is a base-emitter voltage of a forward-biased bipolar transistor. This voltage has a negative TC of about −2.2 mV/° C. and is usually denoted as a Complementary to Absolute Temperature or CTAT voltage. The second voltage which is Proportional to Absolute Temperature, or a PTAT voltage, is formed by amplifying the voltage difference (ΔVbe) of two forward biased base emitter junctions of bipolar transistors operating at different current densities. These type of circuits are well known and further details of their operation is given in Chapter 4 of Analysis and Design of Analog Integrated Circuits, 4th Edition by Gray et al, the contents of which are incorporated herein by reference.
A classical configuration of such a voltage reference circuit is known as a “Brokaw Cell”, an example of which is shown in FIG. 1. First and second transistors Q1, Q2 have their respective collectors coupled to the non-inverting and inverting inputs of an amplifier A1. The bases of the transistors are commonly coupled, and this common node is coupled via a resistor, r5, to the output of the amplifier. This common node of the coupled bases and resistor r5 is coupled via another resistor, r6, to ground. The emitter of Q2 is coupled via a resistor, r1, to a common node with the emitter of transistor Q1. This common node is then coupled via a second resistor, r2, to ground. A feedback loop from the output node of A1 is provided via a resistor, r3, to the collector of Q2, and via a resistor r4 to the collector of Q1.
In FIG. 1, the transistor Q2 is provided with a larger emitter area relative to that of transistor Q1 and as such, the two bipolar transistors Q1 and Q2 operate at different current densities. Across resistor r1 a voltage, ΔVbe, is developed of the form:                               Δ          ⁢                                          ⁢          Vbe                =                                            K              ⁢                                                          ⁢              T                        q                    ⁢          ln          ⁢                                          ⁢                      (            n            )                                              (        1        )            where
k is the Boltzmann constant,
q is the charge on the electron,
T is the operating temperature in Kelvin,
n is the collector current density ratio of the two bipolar transistors.
Usually the two resistors r3 and r4 are equal and the collector current density ratio is given by the ratio of emitter area of Q2 to that of Q1. In order to reduce the reference voltage variation due to the process variation, Q2 may be provided as an array of n transistors, each transistor being of the same area as Q1.
The voltage ΔVbe generates a current, I1, which is also a PTAT current.
The voltage of the common base node of Q1 and Q2 will be:                               V          b                =                              2            ⁢            Δ            ⁢                                                  ⁢            Vbe            *                          r2              r1                                +                      V                          be              ⁢                                                          ⁢              1                                                          (        2        )            
By properly scaling the resistor's ratio and current density the voltage Vb″ is temperature insensitive by the first order, and apart from the curvature which is effected by the base-emitter voltage can be considered as remaining compensated. The voltage Vb is scaled to the amplifier's output as a reference voltage, Vref, by the ratio of r5 to r6:                               V          ref                =                                            (                                                2                  ⁢                  Δ                  ⁢                                                                          ⁢                                      V                    be                                    *                                                            r                      2                                                              r                      1                                                                      +                                  V                                      be                    ⁢                                                                                  ⁢                    1                                                              )                        ⁢                          (                              1                +                                                      r                    5                                                        r                    6                                                              )                                +                                    (                                                                    I                    b                                    ⁡                                      (                                          Q                      1                                        )                                                  +                                                      I                    b                                    ⁡                                      (                                          Q                      2                                        )                                                              )                        ⁢                          r              5                                                          (        3        )            
Here, Ib(Q1) and Ib(Q2) are the base currents of transistors Q1 and Q2.
Although a “Brokaw Cell” is widely used, it still has some drawbacks. The second term in equation 3 represents the error due to the base currents. In order to reduce this error r5 has to be as low as possible. As r5 is reduced, the current extracted from supply voltage via reference voltage increases and this is a drawback. Another drawback is related to the fact that as operating temperature changes the collector-base voltage of the two transistors also changes. As a result of the Early effect (the effect on transistor operation of varying the effective base width due to the application of bias), the currents into the two transistors are affected. Further information on the Early effect may be found on page 15 of the aforementioned 4th Edition of Analysis and Design of Analog Integrated Circuits. 
If the second order effects in the circuit of FIG. 1 are neglected, the amplifier's input offset voltage Voff is reflected into the reference voltage node as:                               V                      ref            -            off                          =                              V            off                    *                      r2            r4                    ⁢                      (                          1              +                              r5                r6                                      )                                              (        4        )            The amplifier's noise is also reflected from input to reference node with the same gain:                     G        =                              r2            r4                    ⁢                      (                          1              +                              r5                r6                                      )                                              (        5        )            
From equation 4 and FIG. 1 it is clear that the easy way to reduce offset and noise sensitivity in a “Brokaw Cell” is to make r4 larger compared to r2. But as r4 is larger, the collector-base voltages of Q1 and Q2 are also larger and Early effect is exaggerated.
The “Brokaw Cell” also suffers, in the same way as all uncompensated reference voltages do, in that it is affected by “curvature” of base-emitter voltage.
The base-emitter voltage of a bipolar transistor, used as a CTAT voltage in bandgap voltage references, and as biased by a PTAT collector current is temperature related as equation 6 shows:                                           V            be                    ⁡                      (            T            )                          =                                            V              G0                        (                          1              -                              T                                  T                  0                                                      )                    +                                    V              be0                        ⁢                          T                              T                0                                              -                                    (                              σ                -                1                            )                        ⁢                          kT              q                        ⁢                          ln              (                              T                                  T                  0                                            )                                                          (        6        )            where:
Vbe(T) is the temperature dependence of the base-emitter voltage for the bipolar transistor at operating temperature,
VBE0 is the base-emitter voltage for the bipolar transistor at a reference temperature,
VG0 is the bandgap voltage or base-emitter voltage at 0 K temperature,
T0 is the reference temperature,
σ is the saturation current temperature exponent (sometimes referred as XTI in computer-aided simulators).
The PTAT voltage developed across r2 in FIG. 1 only compensates for the first two terms in equation 6. The last term, which provides the “curvature” of the order of about 2.5 mV for the industrial temperature range (−40° C. to 85° C.) remains uncompensated and this is gained into the reference voltage by the gain factor G (equation 5).
As the “Brokaw Cell” is well balanced, it is not easy to compensate internally for the “curvature” error. One attempt to compensate for this error is presented in U.S. Pat. No. 5,352,973, co-assigned to the assignee of the present invention, the disclosure of which is incorporated herein by reference. In this US patent, although the “curvature” error is compensated, in this methodology by use of a separate circuit which biases an extra bipolar transistor with constant current, it does require the use of an additional circuit.
Other known examples of band gap reference circuits include those described in U.S. Pat. No. 4,399,398 assigned to the RCA Corporation which describes a voltage reference circuit with feedback which is adapted to control the current flowing between first and second output terminals in response to the reference potential departing from a predetermined value. This circuit is a simple implementation that achieves a reduction of the Early effect. The circuit serves to reduce the base current effect, but at the cost of high power. As a result, this circuit is only suited for relatively high current applications. This can be traced to the fact that the compensation for the base current is effected by operating transistor T1 at a higher current than transistor T2, and as the power is increased the dissipation across RS is also increased. Also, it will be appreciated from an examination of the circuitry that the power supply rejection achieved is relatively modest.
It will be appreciated therefore that although the circuitry described in FIG. 1 has very low offset and noise sensitivity, there is still a need to provide for further reduction in sensitivity to offset and noise.